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Two-channel Kondo phases in coupled quantum dots

Andrew K. Mitchell

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We investigate systems comprising chains and rings of quantum dots, coupled to two metallic leads. Such systems allow to study the competition between orbital and spin degrees of freedom in a nanodevice, and the effect this subtle interplay has on two-channel Kondo (2CK) physics. We demonstrate that a rich range of strongly correlated electron behaviour results, with non-Fermi liquid 2CK phases and non-trivial phase transitions accessible. We employ physical arguments and the numerical renormalization group (NRG) technique to analyse these systems in detail, examining in particular both thermodynamic and dynamical properties. When leads are coupled to either end of a chain of dots, we show that the resulting behaviour on low temperature/energy scales can be understood in terms of simpler paradigmatic quantum `impurity' models. An effective low-energy single-spin 2CK model is derived for all odd-length chains, while the behaviour of even-length chains is related fundamentally to that of the classic `two-impurity Kondo' model. In particular, for small interdot coupling, we show that an effective coupling mediated though incipient single-channel Kondo states drives all odd chains to the 2CK fixed point (FP) on the lowest temperature/energy scales. A theory is also developed to describe a phase transition in even chains. We derive an effective channel-anisotropic 2CK model, which indicates that the critical FP of such models must be the 2CK FP. This physical picture is confirmed using NRG for various chain systems. We also examine the effect of local frustration on 2CK physics in mirror-symmetric ring systems. The importance of geometry and symmetry is demonstrated clearly in the markedly different physical behaviour that arises in systems where two leads are either connected to the same dot, or to neighbouring dots. In the latter case, we show for all odd-membered rings that two distinct 2CK phases, with different ground state parities, arise on tuning the interdot couplings. A frustration-induced phase transition thus occurs, the 2CK phases being separated by a novel critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilise the 2CK FPs, and the critical FP is shown to consist of a free pseudospin together with effective single-channel spin quenching. While connecting both leads to the same dot again results in two parity-distinct phases, a simple level-crossing transition now results due to the symmetry of the setup. The proposed geometry also allows access to a novel ferromagnetically-coupled two-channel local moment phase. Driven by varying the interdot couplings and occurring at the point of inherent magnetic frustration, such transitions in ring structures provide a striking example of the subtle interplay between internal spin and orbital degrees of freedom in coupled quantum dot systems, and the resulting effect on Kondo physics.

DOI
10.5287/ora-qmkybnvb2

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We investigate systems comprising chains and rings of quantum dots, coupled to two metallic leads. Such systems allow to study the competition between orbital and spin degrees of freedom in a nanodevice, and the effect this subtle interplay has on two-channel Kondo (2CK) physics. We demonstrate that a rich range of strongly correlated electron behaviour results, with non-Fermi liquid 2CK phases and non-trivial phase transitions accessible. We employ physical arguments and the numerical renormalization group (NRG) technique to analyse these systems in detail, examining in particular both thermodynamic and dynamical properties. When leads are coupled to either end of a chain of dots, we show that the resulting behaviour on low temperature/energy scales can be understood in terms of simpler paradigmatic quantum impurity' models. An effective low-energy single-spin 2CK model is derived for all odd-length chains, while the behaviour of even-length chains is related fundamentally to that of the classic two-impurity Kondo' model. In particular, for small interdot coupling, we show that an effective coupling mediated though incipient single-channel Kondo states drives all odd chains to the 2CK fixed point (FP) on the lowest temperature/energy scales. A theory is also developed to describe a phase transition in even chains. We derive an effective channel-anisotropic 2CK model, which indicates that the critical FP of such models must be the 2CK FP. This physical picture is confirmed using NRG for various chain systems. We also examine the effect of local frustration on 2CK physics in mirror-symmetric ring systems. The importance of geometry and symmetry is demonstrated clearly in the markedly different physical behaviour that arises in systems where two leads are either connected to the same dot, or to neighbouring dots. In the latter case, we show for all odd-membered rings that two distinct 2CK phases, with different ground state parities, arise on tuning the interdot couplings. A frustration-induced phase transition thus occurs, the 2CK phases being separated by a novel critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilise the 2CK FPs, and the critical FP is shown to consist of a free pseudospin together with effective single-channel spin quenching. While connecting both leads to the same dot again results in two parity-distinct phases, a simple level-crossing transition now results due to the symmetry of the setup. The proposed geometry also allows access to a novel ferromagnetically-coupled two-channel local moment phase. Driven by varying the interdot couplings and occurring at the point of inherent magnetic frustration, such transitions in ring structures provide a striking example of the subtle interplay between internal spin and orbital degrees of freedom in coupled quantum dot systems, and the resulting effect on Kondo physics.
We study theoretically a ring of three quantum dots mutually coupled by antiferromagnetic exchange interactions and tunnel-coupled to two metallic leads: the simplest model in which the consequences of local frustration arising from internal degrees of freedom may be studied within a two-channel environment. Two-channel Kondo (2CK) physics is found to predominate at low energies in the mirror-symmetric models considered, with a residual spin-12\frac{1}{2}21​ overscreened by coupling to both leads. It is however shown that two distinct 2CK phases, with different ground-state parities, arise on tuning the interdot exchange couplings. In consequence a frustration-induced quantum phase transition occurs, the 2CK phases being separated by a quantum critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilize the 2CK fixed points; and the critical fixed point is shown to consist of a free pseudospin together with effective one-channel spin quenching, itself reflecting underlying channel anisotropy in the inherently two-channel system. Numerical renormalization group techniques and physical arguments are used to obtain a detailed understanding of the problem, including study of both thermodynamic and dynamical properties of the system.
We study odd-membered chains of spin-12\frac{1}{2}21​ impurities, with each end connected to its own metallic lead. For antiferromagnetic exchange coupling, universal two-channel Kondo (2CK) physics is shown to arise at low energies. Two overscreening mechanisms are found to occur depending on coupling strength, with distinct signatures in physical properties. For strong interimpurity coupling, a residual chain spin-12\frac{1}{2}21​ moment experiences a renormalized effective coupling to the leads, while in the weak-coupling regime, Kondo coupling is mediated via incipient single-channel Kondo singlet formation. We also investigate models in which the leads are tunnel-coupled to the impurity chain, permitting variable dot filling under applied gate voltages. Effective low-energy models for each regime of filling are derived, and for even fillings where the chain ground state is a spin singlet, an orbital 2CK effect is found to be operative. Provided mirror symmetry is preserved, 2CK physics is shown to be wholly robust to variable dot filling; in particular, the single-particle spectrum at the Fermi level, and hence the low-temperature zero-bias conductance, is always pinned to half-unitarity. We derive a Friedel-Luttinger sum rule and from it show that, in contrast to a Fermi liquid, the Luttinger integral is nonzero and determined solely by the ``excess'' dot charge as controlled by gate voltage. The relevance of the work to real quantum dot devices, where interlead charge-transfer processes fatal to 2CK physics are present, is also discussed. Physical arguments and numerical renormalization-group techniques are used to obtain a detailed understanding of these problems.
We study the low-temperature physics arising in models of a strongly correlated, tunnel-coupled double quantum dot (DQD), particularly the two-impurity Anderson model (2AIM) and the two-impurity Kondo model (2IKM), employing a combination of physical arguments and the Numerical Renormalisation Group. These models exhibit a rich range of Kondo physics. In the regime with essentially one electron on each dot, there is a competition between the Kondo effect and the interdot exchange interaction. This competition gives rise to a quantum phase transition (QPT) between local singlet and Kondo singlet phases in the 2IKM, which becomes a continuous crossover in the 2AIM as a result of the interlead charge transfer present. The 2IKM is known to exhibit two-channel Kondo (2CK) physics at the QPT, and we investigate whether this is also the case for the 2AIM at the crossover. We find that while in principle 2CK physics can be observed in the 2AIM, extremely low temperatures are required, such that it is unlikely that 2CK physics will be observed in an experimental DQD system in the near future. We have studied the effect of a magnetic field on the 2AIM and the 2IKM, finding that both the zero-field QPT in the 2IKM and the zero-field crossover in the 2AIM, persist to finite field. This presents the possibility of observing 2CK physics in an experimental DQD at finite field, but we find that the temperatures required to do so are extremely low. We show that longer even-numbered chains of spins also exhibit QPTs at finite field, and argue that a 2N-spin chain should undergo N QPTs as field is increased (starting deep in the local singlet phase at zero field). We have also carried out a joint theoretical-experimental study of a carbon nanotube based DQD, in collaboration with Dr. Mark Buitelaar et al. The agreement between experimental and theoretical results is good, and the experiments are able to access the crossover present in the 2AIM at finite field. Furthermore, the experiments show the wide range of physics exhibited by DQD systems, and illustrate the utility of such systems in probing correlated electron physics.
The numerical renormalization group is employed to study a double quantum (DQD) dot system consisting of two equivalent single-level dots, each coupled to its own lead and with a mutual capacitive coupling embodied in an interdot interaction U', in addition to the intradot Coulomb interaction U. We focus on the regime with two electrons on the DQD, and the evolution of the system on increasing U'/U. The spin-Kondo effect arising for U'=0 (SU(2) x SU(2)) is found to persist robustly with increasing U'/U, before a rapid but continuous crossover to (a) the SU(4) point U'=U where charge and spin degrees of freedom are entangled and the Kondo scale strongly enhanced; and then (b) a charge-Kondo state, in which a charge-pseudospin is quenched on coupling to the leads/conduction channels. A quantum phase transition of Kosterlitz-Thouless type then occurs from this Fermi liquid, strong coupling (SC) phase, to a broken symmetry, non-Fermi liquid charge ordered (CO) phase at a critical U'_c. Our emphasis in this paper is on the structure, stability and flows between the underlying RG fixed points, on the overall phase diagram in the (U,U')-plane and evolution of the characteristic low-energy Kondo scale inherent to the SC phase; and on static physical properties such as spin- and charge-susceptibilities (staggered and uniform), including universality and scaling behaviour in the strongly correlated regime. Some exact results for associated Wilson ratios are also obtained.
We investigate theoretically the possibility of observing two-channel Kondo (2CK) physics in tunnel-coupled double quantum dots (TCDQDs), at both zero and finite magnetic fields; taking the two-impurity Anderson model as the basic TCDQD model, together with effective low-energy models arising from it by Schrieffer-Wolff transformations to second and third order in the tunnel couplings. The models are studied primarily using Wilson's numerical renormalization group. At zero field our basic conclusion is that while 2CK physics arises in principle provided the system is sufficiently strongly correlated, the temperature window over which it could be observed is much lower than is experimentally feasible. This finding disagrees with recent work on the problem, and we explain why. At finite field, we show that the quantum phase transition known to arise at zero field in the two-impurity Kondo model, with an essentially 2CK quantum critical point, persists at finite fields. This raises the prospect of access to 2CK physics by tuning a magnetic field, although preliminary investigation suggests this to be even less feasible than at zero field.
We investigate theoretically the quantum phase transition (QPT) between the one-channel Kondo (1CK) and two-channel Kondo (2CK) fixed points in a quantum dot coupled to helical edge states of interacting 2D topological insulators (2DTI) with Luttinger parameter 0 < K < 1 . The model was studied by Law et al (2010 Phys. Rev. B 81 041305(R) ), and was mapped onto an anisotropic two-channel Kondo model via bosonization. For K < 1 , the strong coupling 2CK fixed point was argued to be stable for infinitesimally weak tunnelings between the dot and the 2DTI based on a simple scaling dimensional analysis (Law et al 2010 Phys. Rev. B 81 041305(R) . We re-examine this model beyond the bare scaling dimension analysis via a one-loop renormalization group (RG) approach combined with bosonization and re-fermionization techniques near weak-coupling and strong-coupling (2CK) fixed points. We find for a fixed value of K < 1 that the 2CK fixed point can be unstable towards the 1CK fixed point and the system is expected to undergo a quantum phase transition between 1CK and 2CK fixed points with changing Kondo couplings. Our RG approach is controlled near K = 1. In general, this QPT can also occur upon tuning the Luttinger parameter K to a critical value K c smaller than unity ( 0 < K c < 1 ) for fixed Konodo couplings. The QPT in our model comes as a result of the combined Kondo and the helical Luttinger physics in 2DTI, and it serves as the first example of the 1CK-2CK QPT that is accessible by the controlled RG approach. We extract quantum critical and crossover behaviors from various thermodynamical quantities near the transition. Our results are robust against particle-hole asymmetry for 1 2 < K < 1 .